A property of a simply ordered set
K. W. Folley · Bulletin of the American Mathematical Society · 1940
Sierpinski 2 has shown that the set of real numbers of the interval (0, 1) may be decomposed into c disjoint subsets, each of power less than c, and such that the sum of every c of these subsets has at least one point in common with every perfect subset of the interval.The object of the present paper is to show that the same method of proof may be used to prove an analogous theorem concerning a more general type of set. DEFINITIONS. A simply ordered set M is a set such that if any two of its elements are given it is known which one precedes.A subset of M is said to be cofinal (coinitial) with M if no element of M follows (precedes) all the elements of the subset.An 7) a subset of M is one which is neither cofinal nor coinitial with any subset of M of power less than N a and which contains no pair of neighboring subsets both of which have power less than \& a .